Question

You are given the following pieces of information: 12 ∫ 57

f (x) dx = −30, 27 ∫ 57 f (x) dx = −18, 45 ∫ 12 f (x) dx =
35. What is the value of 45 ∫ 2

Answer 1

Using properties of definite integrals and given integral values, the value of 45 ∫ 27 f(x) dx is calculated to be -13.

Step-by-step explanation:

To find the value of 45 ∫ 27 f(x) dx, we need to use the given information about the integrals of the function f(x) over various intervals:

• 12 ∫ 57 f(x) dx = −30
• 27 ∫ 57 f(x) dx = −18
• 45 ∫ 12 f(x) dx = 35

The properties of definite integrals allow us to deduce that:

1. By adding the first two given integrals, we get 12 ∫ 57 f(x) dx + 27 ∫ 57 f(x) dx = 12 ∫ 27 f(x) dx, which simplifies to −30 + (−18) = 12 ∫ 27 f(x) dx.
2. To find the value of 45 ∫ 27 f(x) dx, we need to subtract the integral 45 ∫ 12 f(x) dx from the result of 12 ∫ 27 f(x) dx.
3. Therefore, we calculate 45 ∫ 27 f(x) dx by 12 ∫ 27 f(x) dx − (45 ∫ 12 f(x) dx), which equals to (−30 + −18) − 35 = −48 − 35 = −13.

Hence, the value of 45 ∫ 27 f(x) dx is −13.